# 不动点迭代和优化方法

### Fixed Point Iteration

If a function $f$ defined on the real line with real values is Lipschitz continuous with Lipschitz constant $L<1$, then this function has precisely one fixed point, and the fixed-point iteration converges towards that fixed point for any initial guess $x_{0}$. 此定理可以推广到任意的度量空间，只要满足映射是压缩映射，有兴趣的可以参看Banach fixed-point theorem

### Unconstrainded Optimization

$\alpha f'(x)=0$ $x_1 = x_0-\alpha f'(x_0)$ $x_2 = x_1-\alpha f'(x_1)$ $\cdots$ $x_n = x_{n-1}- \alpha f'(x_{n-1})$